Split lake model lecture into two consecutive lectures (#665)

* Split lake model lecture into two consecutive lectures

Split the lake_model.md lecture into two parts to improve readability and organization:

1. **lake_model.md** - Basic lake model with exogenous transition rates
   - Covers the foundational lake model theory
   - Aggregate and rate dynamics
   - Individual worker dynamics and ergodicity
   - Includes exercises that use only basic lake model functions

2. **endogenous_lake.md** - Extension with endogenous job finding rate
   - Makes the job finding rate endogenous using McCall search model
   - Fiscal policy analysis with unemployment insurance
   - Welfare optimization

**Key changes:**
- Created new endogenous_lake.md file with content from original line 624 onwards
- Updated lake_model.md to end before the endogenous section
- Added clear introduction to endogenous_lake.md indicating it's a continuation
- Moved exercises (model_ex2, model_ex3) to end of lake_model.md (they only use basic functions)
- Consolidated parameter source citations in lake_model.md where defaults are defined
- Removed redundant parameter value listings in exercises
- Updated plot colors to cycle through matplotlib's default colors (C0 for lines, C1 for hlines)
- Updated _toc.yml to include both lectures consecutively

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

* Improve endogenous_lake.md: Split code blocks and add descriptions

Reorganized the endogenous lake model lecture for better readability and comprehension.

**Changes:**
- Split large monolithic code blocks into 12 smaller, focused blocks
- Added descriptive sentences before each code block explaining its purpose
- Separated function definitions from their usage and plotting code
- Added new subsection "Computing optimal unemployment insurance"

**Code block breakdown:**
1. Wage distribution function definition
2. Wage distribution creation and visualization
3. Utility function and McCall model data structure
4. Bellman equation operator
5. Value function iteration solver
6. Lake model functions (from previous lecture)
7. Economy parameters container
8. Function to compute optimal worker quantities
9. Function to compute steady state outcomes
10. Function to find balanced budget tax rate
11. Computation loop across policy range
12. Results visualization

Each block now has a clear explanation of what it does and how it fits into the overall analysis, making the lecture more pedagogically effective.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

* Add exercise to endogenous_lake.md lecture

Added an exercise exploring how the welfare-maximizing level of unemployment
compensation varies with the job separation rate.

**Exercise:**
- Computes optimal unemployment compensation for different separation rates
- Uses brute force search over c_vec to find welfare-maximizing c
- Plots the relationship between α and optimal c
- Includes economic interpretation of results

**Solution shows:**
- As separation rate increases, optimal unemployment insurance increases
- Makes intuitive sense: more frequent job loss → higher value of insurance
- Demonstrates practical application of the welfare optimization framework

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

* misc

* Fix unstable exercise solution in endogenous_lake.md

The exercise solution was producing an unstable, noisy relationship between
separation rate and optimal unemployment compensation due to extremely flat
welfare function near its maximum.

Changes:
- Expanded α range from [0.01, 0.025] to [0.01, 0.04] with fewer points (8 vs 15)
- Increased c grid resolution from 40 to 150 points
- Implemented centroid method: compute weighted average of near-optimal c values
  instead of using argmax, which is unstable on flat functions
- Updated solution text to correctly state that optimal c decreases (not increases)
  with separation rate, with proper economic explanation

Result: Clean, monotonically decreasing relationship from c=68.88 to c=60.72
as α increases from 0.01 to 0.04.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

* Update lectures/endogenous_lake.md

* Fix syntax error in endogenous_lake.md

Removed incomplete statement `model.` on line 489 that was causing a syntax error when converting to Python with jupytext.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude <noreply@anthropic.com>

---------

Co-authored-by: Claude <noreply@anthropic.com>
Co-authored-by: Matt McKay <mmcky@users.noreply.github.com>

Author: 3 people

lectures/_toc.yml

@@ -101,6 +101,7 @@ parts:
   numbered: true
   chapters:
   - file: lake_model
+  - file: endogenous_lake
   - file: rational_expectations
   - file: re_with_feedback
   - file: markov_perf